IMPLICITIZATION OF SURFACES IN ℙ<sup>3</sup> IN THE PRESENCE OF BASE POINTS

Busé, Laurent; Cox, David A.; DʼAndrea, Carlos

doi:10.1142/s0219498803000489

Cited by 81 publications

(71 citation statements)

References 23 publications

(41 reference statements)

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“…T = P 2 , and [BCD03] does the same in the presence of base points. In [AHW05], square matrix representations of bihomogeneous parametrizations, i.e.…”

Section: Introductionmentioning

confidence: 89%

Matrix representations for toric parametrizations

Botbol¹,

Dickenstein²,

Dohm³

2009

Computer Aided Geometric Design

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In this paper we show that a surface in P 3 parametrized over a 2-dimensional toric variety T can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This constitutes a direct generalization of the corresponding result over P 2 established in [BJ03] and [BC05]. Exploiting the sparse structure of the parametrization, we obtain significantly smaller matrices than in the homogeneous case and the method becomes applicable to parametrizations for which it previously failed. We also treat the important case T = P 1 × P 1 in detail and give numerous examples.

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“…T = P 2 , and [BCD03] does the same in the presence of base points. In [AHW05], square matrix representations of bihomogeneous parametrizations, i.e.…”

Section: Introductionmentioning

confidence: 89%

Matrix representations for toric parametrizations

Botbol¹,

Dickenstein²,

Dohm³

2009

Computer Aided Geometric Design

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“…Several other authors have investigated the structure of the resultant of a μ-basis of a rational surface using subtle algebraic techniques; see, for example, Theorem 4.1 of [3] and Proposition 7 of [2]. For a rational surface P(s, t, u) with parameters in complex projective space P 2 (C), the authors of [3] prove that under certain assumptions, the resultant of a μ-basis of P(s, t) is a power of the implicit equation.…”

Section: Implicit Equationmentioning

confidence: 99%

Strong $\mu$-Bases for Rational Tensor Product Surfaces and Extraneous Factors Associated to Bad Base Points and Anomalies at Infinity

Shen¹,

Goldman²

2017

SIAM J. Appl. Algebra Geometry

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Abstract. We investigate conditions under which the resultant of a µ-basis for a rational tensor product surface is the implicit equation of the surface without any extraneous factors. In this case, we also derive a formula for the implicit degree of the rational surface based only on the bidegree of the rational parametrization and the bidegrees of the elements of the µ-basis without any knowledge of the number or multiplicities of the base points, assuming only that all the base points are local complete intersections. We conclude that in this case the implicit degree of a rational surface of bidegree (m, n) is at most mn, so the rational surface must have at least mn base points counting multiplicity. When the resultant of a µ-basis generates extraneous factors, we show how to predict and compute these extraneous factors from either the existence of bad base points or anomalies occurring in the parametrization at infinity. Examples are provided to flesh out the theory.

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“…However, the same proof can be applied without modifications to our setting: The key property used in the proof is the fact that the canonical map A 1 ⊗ A n → A n+1 is surjective and this is also valid for A = K[X]/(X 1 X 4 − X 2 X 3 ). Moreover, by (4) we have that ann K[T ] (Sym A (I) ν ) = 0 for ν ≫ 0 if and only if P is locally generated by at most 3 equations, and in this case it is clear that it is contained in ker(h). Finally, if P is locally defined by at most 2 equations, meaning that P is locally a complete intersection, then I is of linear type outside V (m) (use for instance [5,Propositions 4.1 and 4.5]) which shows the last claimed equality as proven in [5,Proposition 5.1].…”

Section: Acyclicity Criterionmentioning

confidence: 99%